Arithmetic Sequences & Series
TEACHER NOTES
About the Lesson
Students use formulas to find the differences of the consecutive
terms, plot a scatter plot of each sequence, and determine that
sequences with common differences (called arithmetic
sequences) have scatter plots whose points form a straight line.
They then learn how to find the sum of the first n terms of the
related series. As a result, students will:
 Given several terms of a sequence, write an algebraic
th
expression that generates the n term.
 Graph the first n terms of a sequence.
Tech Tips:
 Derive and apply a formula for the first n terms of an arithmetic

This activity includes screen
captures taken from the TI-84
sequence.
Plus CE. It is also appropriate for
use with the rest of the TI-84 Plus
Vocabulary
family. Slight variations to these
 arithmetic sequence
directions may be required if
 common difference
using other calculator models.
 explicit formula

 arithmetic series
Watch for additional Tech Tips
throughout the activity for the
specific technology you are using.

Teacher Preparation and Notes
 It would be beneficial for students to clear all lists and
functions. Press y à and select ClearAllLists. Press o,
move to any equation that is defined and press ‘.
 Students should begin this activity knowing that a sequence is
an ordered list of numbers that follows a pattern and that a
series is an indicated sum of a sequence. For example, 1, 2,
3, 4 is a sequence and 1 + 2 + 3 + 4 is a series.
Access free tutorials at
http://education.ti.com/calculators/
pd/US/Online-Learning/Tutorials

Any required calculator files can
be distributed to students via
handheld-to-handheld transfer.
Lesson Files:
 Arithmetic_Sequences_Series_
Student.pdf
Activity Materials
 Arithmetic_Sequences_Series_
 Compatible TI Technologies:
Student.doc
TI-84 Plus*
TI-84 Plus Silver Edition*
TI-84 Plus C Silver Edition
TI-84 Plus CE
* with the latest operating system (2.55MP) featuring MathPrint TM functionality.
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Arithmetic Sequences & Series
TEACHER NOTES
Part 1 – Sequences and Scatter Plots
Students are to create a new data table by pressing … and
choosing Option 1: Edit under the EDIT menu. Instruct them
to copy the data shown into L1 and L2, find the differences
between consecutive terms of the sequence in L2, and record
them in L3. They can either use the list/row numbers, L2(2)–
L2(1), or the actual numbers, such as 8.75–7.5.
Students will then enter the terms of a sequence into L4, find
the consecutive differences for the L4 sequence, and record
them in L5.
For each sequence, graph the ordered pairs formed by the
domain (the natural numbers listed in L1) and the terms of the
sequence.
Students will plot L2 in Plot1 and L4 in Plot2. The Xlist for
both plots should be L1. Have them press q and select
9:ZoomStat to view the plots. If they have trouble viewing
both plots together, have them turn off one plot and then the
other.
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Arithmetic Sequences & Series
TEACHER NOTES
1. For each sequence, write the differences between the consecutive terms and give a description of the
scatter plot.
a. Sequence L2
Answer: 1.25, 1.25, 1.25, 1.25, 1.25. Possible answer: The points of the scatter plot form a straight
line that slants up to the right.
b. Sequence L4
Answer: 3, 5, 8, 13, 21. Possible answer: The points of the scatter plot form a curve.
c. Study the graphs and the differences you found in L3 and L5. Make a conjecture.
Answer: Students should make conjectures about the scatter plot of a sequence and the
differences between the consecutive terms. They should conjecture that for sequences
with a common difference, the points form a straight line.
Now students are to clear the data from L2, L3, L4, and L5 but
leave the numbers in L1. To do this, they can arrow up to the
top of each list and press ‘ Í.
After entering the sequences shown at the right into L2 and
L4, they will find the differences between consecutive terms,
recording them in L3 and L5.
Students can press q and select 9:ZoomStat to view the
plots of the sequences.
For each sequence, students are to write the differences
between the consecutive terms and give a description of the
scatter plot.
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Arithmetic Sequences & Series
TEACHER NOTES
2. For each sequence, write the differences between the consecutive terms and give a description of the
scatter plot.
a. Sequence L2
Answer: –5, –5, –5, –5, –5. Possible answer: The points of the scatter plot form a straight line that
slants down to the right.
b. Sequence L4
Answer: 1, 2, 3, 4, 5. Possible answer: The points of the scatter plot form a curve.
c. How do your observations affect your conjecture about the scatter plot of a
sequence and the differences between the consecutive terms? Explain.
Answer: Students should find that the new data reinforces their conjecture that for sequences with
a common difference, the points form a straight line.
Part 2 – Explicit Formulas and Sums
Students are shown the general explicit formula for an
arithmetic sequence.
un = u1 + (n -1)d
They are to generate a sequence in L2 to display the first 30
terms of un = 7.5 + (n -1) i1.25 . They will use the sequence
command by pressing y … [list], arrow over to the OPS
menu and select seq(.
Inside the parentheses, 7.5+(N–1)*1.25 is the explicit formula
for this sequence (replace u1 with 7.5 and d with 1.25 in
un = u1 + (n -1)d , N is the variable, 1 is the first term to
display, and 30 is the number of terms to display.
Note: N is selected by using the ƒ key.
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Arithmetic Sequences & Series
TEACHER NOTES
Now students are to simplify the formula
un = 7.5 + (n -1) i1.25 by distributing and combining like
terms: un = 1.25n + 6.25 . They need to use this formula and
the sequence command to generate 30 terms of this
sequence in L3.
Note: It is not necessary that students number L1 all the way
to 30.
3. Simplify the formula un = 7.5 + (n -1) i1.25 by distributing and combining like terms. Use this formula in
the sequence command to generate 30 terms of this sequence in L3.
What do you notice about the terms in L2 and L3?
Answer: The terms that appear should be the same as in L2.
Part 3 – Practice Finding the Sum of a Series
Now students are to find the sum of the first 30 terms of the
sequence from problem 2. The expression consisting of
summing the terms in a sequence is called a series.
On the Home screen students are to enter sum(L2). The sum
command can be entered by pressing y … [list], arrow
over to the MATH menu and select sum(.
4. What is the sum of the series in L2?
Answer: 768.75
Now they are to find the sum of the first 80 terms of the
sequence below, using the Lists feature and the sum()
command.
62, 67, 72, 77, 82…
Students will need to clear their L2 and L3 lists and use the
same procedure as before.
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Arithmetic Sequences & Series
TEACHER NOTES
5. Now, let’s look at another sequence. Find the sum of the
first 80 terms of the sequence below, using the Lists
feature and the sum() command.
62, 67, 72, 77, 82…
a. Find the explicit formula for this sequence in simplified
form.
Answer: 5n + 57
b. What is the sum of the first 80 terms?
Answer: 20,760
Extension
As an extension, explain to students that the sum of the first n terms of an arithmetic series can be found
by multiplying the number of terms, n, by the average of the first and last terms.
Have students use their calculator to show that this holds true for the sums found in Parts 2 and 3 of this
activity.
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